Alan Deardor¤’s Contributions on Trade and Growth
Gene M. Grossman
Princeton University
September 2009
To be presented at Comparative Advantage, Economic Growth, and the Gains from Trade and Globalization: A
Festschrift in Honor of Alan V. Deardor¤, sponsored by the Department of Economics and the Ford School of Public
Policy at the University of Michigan. The festschrift will be held at the University of Michigan on October 2-3, 2009.
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Introduction
It has been a pleasure to re-read some of Alan Deardor¤’s papers on trade and growth, and to read
others for the …rst time. These papers are typical of Alan’s work; they artfully tease new insights
on important issues from simple, familiar models.
It is …tting to begin my review with “A Geometry of Growth and Trade.” Alan loves diagrams
and given his skill in developing them, it is easy to see why. His diagrammatic analyses are as
incisive as any algebraic treatment and more pleasing to the eye.
Deardor¤ (1974) provides a simple geometric tool for analyzing trade and growth in a small
open Solovian economy. Consider a small economy that produces a single consumer good and a
single investment good from two factors of production, capital and labor. For simplicity, take the
labor force as …xed.1 Suppose that households save a constant fraction s of income and that capital
depreciates at constant rate . In Figure 1, R (p; K) represents the revenue or national-product
function for an economy facing world relative price p of the consumer good and having a stock of
capital K. As is well known from microeconomic theory, a competitive economy maximizes the
value of national output given prices. Therefore, the equilibrium allocation of capital and labor is
such that R (p; K) is the economy’s national income. And, as is well known from trade theory, a
two-sector economy facing a given relative price will specialize in producing the labor-intensive good
when its capital stock is small, specialize in producing the capital-intensive good when its capital
stock is large, and will produce both goods in an intermediate “cone of diversi…cation.”Therefore,
the national-product function has two curved portions that depict the diminishing returns to capital
when only one good is being produced, and a linear segment whose slope represents the constant
marginal product of capital within the diversi…cation cone.
With a constant savings propensity, sR (p; K) represents national savings, which fully …nances
national investment in an economy that cannot borrow or lend internationally. The dashed line,
K, represents aggregate deprecation, and the gap between the two is net investment. The capital
stock grows when sR (p; K) exceeds K and shrinks when it is smaller. The intersection of the two
curves depicts the steady state. The diagram readily yields predictions about the evolution of the
trade pattern and about the e¤ects of changes in savings behavior or world prices on the growth
path and the ultimate steady state. For example, an increase in the savings propensity leads to a
larger steady-state stock of capital, more output of the capital-intensive good, and greater exports
(or fewer imports) of that good.
Deardor¤ (1974) employs the simple assumption that national savings is a constant fraction of
national income. But a similar tool to his can be used to study transitional dynamics in an open
economy with optimal savings behavior. Suppose the representative consumer allocates spending
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The original paper incorporates constant, exogenous population growth by treating the capital-to-labor ratio as
the state variable.
1
δK
Savings
R(p,K)
sR(p,K)
.
K=sR(p,K)-δK
K
Figure 1: Two-Sector Solovian Trade and Growth
to maximize an intertemporal utility function of the form2
Z
1
e
(
t)
log c ( ) d
t
where c ( ) is consumption at time
and
is a constant discount factor. Then, as is well known,
the Euler equation implies
E_
=i
E
;
(1)
i.e., the households adjusts expenditure E = pc so that the rate of growth of spending is equal
to the di¤erence between the interest rate and the discount rate. Capital is the only asset in the
model and the investment good is numeraire, so the rate of interest i must equal the real return
on capital net of depreciation; i.e., i = r (p; K)
, where r (p; K) = @R=@K is the rental rate on
capital. Finally, savings— which is the di¤erence between national income and spending— …nances
net investment as in Deardor¤ (1974):
K_ = R(p; K)
E
K.
(2)
Equations (1) and (2) can be used to construct a phase diagram that bears a strong similarity
to the Deardor¤ diagram. In Figure 2, the K_ = 0 curve represents the equation E = R (p; K) K.
The qualitative properties of this curve follow from those of the national-product function, which
Deardor¤ has discussed. The E_ = 0 curve depicts the values of K and E such that r (p; K) = + .
Two comments are in order about this curve. First, the level of spending E does not a¤ect anything
in this equation, so the “curve” is in fact a vertical line. Second, the factor-price equalization
theorem implies a single value of r for all values of K in the diversi…cation cone. If this value of r
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It is easy to handle the case of a constant elasticity of intertemporal substitution; the text describes the special
case where this elasticity is equal to one.
2
does not happen to equal
+ , as it generally will not for arbitrary p, then the E_ = 0 curve will
not be located at any value of K in the diversi…cation cone.
S
E
E=0
S
K=0
K
Figure 2: Two-Sector Neoclassical Trade and Growth
The …gure shows the “arrows”of adjustment that apply to either side of each curve. It is readily
seen that the system exhibits saddle-path stability. For given initial K, there is a single value of
the initial level of spending that avoids K ! 0 and K ! 1 as time progresses. This initial value
of E is the only one that allows for satisfaction of the intertemporal budget constraint and the
transversality condition for optimization of lifetime utility. The economy approaches the steady
state along the stable arm, denoted by SS in the …gure.
The …gure can be used much as Deardor¤’s original diagram. It is easy to track the evolution
of the trade pattern and to perform comparative statics with respect to changes in the discount
rate or the international relative price. One conclusion di¤ers from that in the Solovian world: a
small open economy with optimal savings and a constant discount rate is quite unlikely to remain
incompletely specialized in the long run.
I see several important substantive themes in Alan’s other writings on trade and growth that
have gained traction in the more recent literature. First, trade may be harmful to a growing
economy in some circumstances. Second, and related, an open economy may be trapped in poverty
in a world with multiple steady states. Third, growth may be sustained by trade in a neoclassical
economy that would be doomed to stagnation if it remained closed to international exchange. I
take up each point in turn.
In Deardor¤ (1973), Alan analyzed the e¤ects of trade on per capita consumption in steady state
and in the approach to steady state. He considered an open economy capable of producing a single
consumer good and a single investment good that saves a constant proportion of its income, as in
Deardor¤ (1974). He proved, for example, that a small economy that saves in excess of the golden
rule savings ratio will experience a reduction in steady-state consumption if the world relative price
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of the consumption good is a bit above its autarky price.3 If its savings rate falls short of the
golden rule savings ratio, then steady-state consumption falls with an opening to trade if the world
relative price of the consumption good is a bit below the autarky price. Intuitively, steady-state
consumption increases if a country starts in autarky with less investment than is needed on the
margin to maintain a unit of capital and it exports the investment good, or if it starts with more
capital than is needed on the margin to maintain a unit of capital and it imports the capital good.
The paper also considers the e¤ects of trade on per capita consumption in the short run; that
is, in the moments after an opening of trade. If trade causes the relative price of the consumption
good to rise, thereby generating exports of this good, then per capita consumption will initially
fall. If the relative price moves in the opposite direction and the country exports the investment
good, per capita consumption will rise.
This analysis su¤ers from two shortcomings. First, as Alan himself recognized, it is not typically
optimal for households to consume a constant fraction of their current income. Alan thus relates
his …ndings to the theory of the second best. But convincing second-best arguments that question
the gains from trade usually refer to realistic market failures, not to empirically unsupported and
somewhat arbitrary assumptions of sub-optimal behavior. Second, it is impossible to evaluate
the welfare e¤ects of an opening of trade that induces short-run gains and long-run losses (or
vice versa) without reference to some intertemporal utility function. The Solow model o¤ers no
such utility function and therefore no metric for welfare comparisons. Subsequent to Deardor¤
(1973), Samuelson (1975) and Smith (1979, 1984) showed rigorously that a neoclassical economy
with well-functioning markets always gains from trade when savings decisions derive from utility
maximization, no matter what is the form of its intertemporal utility function.
But the possibility of losses from trade has been emphasized in recent writings on trade and
growth in non-neoclassical settings. Most common are models in which growth is driven by local
knowledge spillovers. These can be spillovers in human capital accumulation, as in Lucas (1988)
and Stokey (1991), or spillovers in the R&D process, as in Young (1993), Feenstra (1996) and
Grossman and Helpman (1991, ch. 8; 1994).
The idea is quite simple. Consider a two-sector economy, one in which technology is static and
another in which technology can improve with the accumulation of human capital or knowledge.
There are spillovers in the accumulation process that o¤set the private diminishing returns, so
that growth can be sustained. In the autarky equilibrium, the economy is incompletely specialized
and the ongoing activity in the dynamic sector ensures the continuing accumulation of knowledge.
Although the autarky allocation of resources to the dynamic sector will be sub-optimally small in
the absence of a Pigouvian subsidy that addresses the externality, growth is sustained. Now open
the economy to trade and suppose that the country has a comparative disadvantage in the dynamic
sector, either due to its relatively unsuitable resource endowments or to an initial disadvantage of
history that gives its trade partner a technological head start. In either case, the opening of
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The golden rule savings rate is the value of s that generates the golden rule capital-to-labor ratio as its steady
state. The golden rule capital-to-labor ratio, in turn, is the value of K=L that makes the marginal product of capital
equal to the sum of the population growth rate and the depreciation rate.
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δK
Savings
R(p,K)
sw(p,K)
.
K=sw(p,K)-δK
K1
K2 K3
K4
K
Figure 3: Multiple Steady States
trade will cause the country to specialize relatively and perhaps fully in the industry with a static
technology. Its growth in output will slow and perhaps cease as a result of trade. Even so, trade
may be gainful, since the country will be able to import the good produced in the dynamic sector
at a lower price than in autarky. But it is easy to construct examples where trade is harmful in
these circumstances, for reasons to do with the theory of the second best. Inasmuch as the autarky
equilibrium entails too little production of the dynamic good, if trade drives resources out of this
industry it can generate losses by exacerbating a pre-existing distortion. Of course, losses from
trade would not be possible if the opening of trade were accompanied by an appropriate Pigouvian
subsidy to the externality-generating activity.
Deardor¤ (2001) makes a related but di¤erent point, drawing on work by Galor (1996). He considers a neoclassical economy that potentially can produce three goods, one investment good and
two consumption goods, with capital and labor. The industries di¤er in their factor intensities, so
there are two cones of diversi…cation. Following Galor, he supposes that savings are approximately
proportional to the wage bill, perhaps because their are overlapping generations and each generation earns capital income only in the last period of life, when it consumes all. Figure 3— analogous
to Figure 1— shows the aggregate savings for this economy as a function of its capital stock. For
low values of K, the economy is incompletely specialized and the wage bill rises as capital is accumulated. The region between K1 and K2 represents the …rst diversi…cation cone, where the country
produces the two least capital-intensive goods. In this range, factors prices are insensitive to factor
endowments, so the wage bill is constant, as is aggregate savings. Further capital accumulation
leads to a range of specialization in the intermediate good, again with a rising wage bill, and then
a second region of diversi…cation, for capital stocks between K3 and K4 . Finally, if capital were
to accumulate beyond K4 , the economy would specialize in producing the most capital-intensive
good, and savings would rise with K.
As is clear from the …gure, this economy can have three distinct steady states, the …rst and third
of which are locally stable. In other words, an economy such as this can get stuck in a “poverty
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trap.”If it starts with a capital stock less than K1 and trades with an otherwise identical economy
that begins with more capital, then it will accumulate capital until it reaches the …rst steady state,
whereupon its growth will cease at a relatively low level of per capita income.4 Di¤erences in initial
conditions are su¢ cient to generate long-run di¤erences in income and welfare for countries that
are otherwise the same in terms of their technologies and savings behavior.
This is an interesting …nding inasmuch as multiple equilibria are not common in neoclassical
growth models. The allusion to an overlapping-generations setup is intriguing, as it seems possible
to have a poverty trap together with fully optimal savings behavior and convex technologies.
The potential for multiple equilibria among identical countries also arises in non-neoclassical
settings. In fact, this possibility is quite natural in models with static or dynamic increasing
returns to scale. Azariadis and Drazen (1990) o¤er an example with multiple steady states even
in the closed economy. They consider an economy with “threshold externalities”; i.e., externalities
generated by human capital that are relatively weak when the stock of human capital is small, but
grow stronger as the skill level increases. In such a setting, an economy can get trapped in an
equilibrium with low skills, despite the fact that it has the potential for sustained growth were it
to somehow manage to escape the trap.5 Young (1993), Feenstra (1996), Grossman and Helpman
(1991, ch.8) and others show the possibility for trade to generate permanent income (or growth rate)
di¤erences between otherwise identical countries that di¤er in their initial levels of technological
development. They consider an international equilibrium with trade between countries that di¤er
only in their initial levels of technological development. Were the countries to remain closed, they
would converge upon similar long-run steady state growth paths. However, with trade, the leading
country gains an initial advantage in the more dynamic sector, and the advantage is perpetuated
(or even extended) over time.
In neoclassical growth models, diminishing returns to capital typically spell the stagnation of
growth in per capita incomes as increases in the capital-to-labor ratio drive down the marginal return
to investment. Growth need not peter out however, as Solow (1956) himself noted, if the return
to capital is bounded from below. This idea of bounded long-run returns to capital has featured
prominently in a branch of the literature on endogenous growth, where models with such features
have been termed “AK models.”See, for example, Jones and Manuelli (1990) for an application to
the open economy. However, many have questioned the plausibility of the assumption that marginal
returns will be bounded from below as the capital-to-labor ratio grows inde…nitely.
Deardor¤ (1994) was one of the …rst to point out that growth can be sustained in some circumstances in an open neoclassical economy even if the technology does not admit a lower bound
on the return to capital. His explanation relies on cross-country di¤erences in population growth
rates and the opportunities a¤orded by international investment.6
4
If the economy is large, the equilibrium prices may be changing in the transition, in which case the national income
and aggregate savings curves will be shifting about. Nonetheless, the point remains that there can be multiple steady
states and a stable equilibrium at a low level of national income. With more than three goods, the number steady
states can increase.
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One way out might be with a “big push,” as emphasized by Murphy, Shleifer and Vishny (1989).
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Deardor¤ (1999) uses a similar framework to study the evolution of international inequality in per capita incomes.
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Consider a pair of Solovian economies that each produce a single good. Let the population
growth rates and the savings rates be exogenous and country speci…c. Alan refers to the country
with the slower population growth as the North and the country with the larger population growth
as the South. Technologies are such that both the North and the South with approach constant
steady-levels of per capita income in their autarkic equilibria. But suppose now that the North
can invest its savings in the South. Then, for some savings rates, it avoid the otherwise inevitable
rise in the capital-to-labor ratio by making use of the ever-larger Southern labor force. When the
North has slower population growth and a su¢ ciently large savings rate, per capita income in the
North grows forever. For even larger Northern savings rates, national income growth in the North
matches the rate of population growth in the South, and residents over the North come to own a
signi…cant portion of the world’s capital despite being a vanishing fraction of the world’s labor force.
Deardor¤ (1994) observes that, in circumstances in which the North enjoys sustained growth in per
capita income but national income growth less than the rate of population growth in the South,
a change in the North’s savings propensity will change its long-run growth, just as in models of
endogenous growth. In short, Alan points out that, by investing abroad, a relatively small country
(in terms of population) can escape diminishing returns at home.
This argument bears a family resemblance to a related point made by Ventura (1997). He
considers a small, Heckscher-Ohlin economy that trades freely at …xed prices. The economy has
two sectors with diminishing (and non-bounded) returns to capital in each, and savings derived
from intertemporal utility maximization. Without trade, this economy would approach a steady
state. With trade, it will do so as well, as was illustrated in Figure 2. But, Ventura points out,
the country will experience a potentially-long growth phase when its endowments are within the
diversi…cation cone during which the return to capital will remain constant. The constancy of
returns to capital re‡ects the factor-price equalization theorem applies during this phase. But as
long as the country remains incompletely specialized, it is as if it had access to an AK technology.
So the growth experience for a long time might mimic that which would be predicted by such a
model. And, of course, changes in policy and in savings behavior will alter the growth rate during
this episode.
Deardor¤’s (1994) story of sustained growth is an interesting one that deserves further attention
and development. To me, it seems to beg for the endogenization of population growth. Can
populations diverge forever? Might the South of Deardor¤’s model experience a demographic
transition at some stage? Or might trade postpone or even prevent such a transition? Going
further, can we justify sustained di¤erences in savings behavior? Are these di¤erences “cultural”
or do they re‡ect the growth experience? There is much to be done with endogenous preferences
and endogenous procreation in models of trade and growth.7
Let me end this review where I began. I have long been a big fan of Alan Deardor¤, whose work
is always clean, crisp and elegant. His papers on trade and growth complement the many other
areas to which he has contributed, including his brilliant work on comparative advantage and his
7
See Galor and Weil (2000) for a very interesting contribution of this sort.
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very useful applied research on trade policy. I am happy to be part of this celebration and look
forward to his continued productivity for many years to come!
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References
[1] Azariadis, Costas and Drazen, Allan (1990), “Threshold Externalities in Economic Development,” Quarterly Journal of Economics 105:2, 501-526.
[2] Deardor¤, Alan V. (1973), “The Gains from Trade In and Out of Steady-State Growth,”
Oxford Economic Papers 25:3, 173-191.
[3] Deardor¤, Alan V. (1974), “A Geometry of Trade and Growth,” Canadian Journal of Economics 7:2, 295-306.
[4] Deardor¤, Alan V. (1994), “Growth and International Investment with Diverging Populations,”
Oxford Economic Papers 46:3, 477-491.
[5] Deardor¤, Alan V. (1999), “Diverging Populations and Endogenous Growth in a Model of
Meaningless Trade,” Review of International Economics 7:3, 359-377.
[6] Deardor¤, Alan V. (2001), “Rich and Poor Countries in Neoclassical Trade and Growth,”
Economic Journal 111:470, 277-294.
[7] Feenstra, Robert C. (1996), “Trade and Uneven Growth,”Journal of Development Economics
49:1, 229-256.
[8] Galor, Oded (1996), “Convergence? Inferences from Theoretical Models,” Economic Journal
106:437, 1056-1069.
[9] Galor, Oded and Weil, David N. (2000), “Population, Technology and Growth: From Malthusian Stagnation to the Demographic Transition and Beyond,” American Economic Review
90:4, 806-828.
[10] Grossman, Gene M. and Helpman, Elhanan (1990), “Trade, Innovation, and Growth,” American Economic Review (Papers and Proceedings) 80:2, 86-91.
[11] Grossman, Gene M. and Helpman, Elhanan (1991), Innovation and Growth in the Global
Economy, MIT Press:Cambridge.
[12] Jones, Larry E. and Manuelli, Rodolfo (1990), "A Convex Model of Equilibrium Growth:
Theory and Policy Implications,” Journal of Political Economy 98:5, 1008-1038.
[13] Lucas, Robert E., Jr. (1988), “On the Mechanics of Economic Development,”Journal of Monetary Economics 22:1, 3-42.
[14] Murphy, Kevin M., Shleifer, Andrei, and Vishny, Robert W. (1989), “Industrialization and the
Big Push,” Journal of Political Economy 97:5, 1003-1026.
[15] Samuelson, Paul A. (1975), Trade Pattern Reversals in Time-Phased Ricardian Systems and
Intertemporal E¢ ciency,” Journal of International Economics 5:4, 309-363.
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[16] Smith, Alasdair (1979), “Intertemporal Gains from Trade,” Journal of International Economics 9:2, 239-248.
[17] Smith, Alasdair (1984), “Capital Theory and Trade Theory,” in R.W. Jones and P.B.Kenen,
eds., Handbook of International Economics, vol. 1, North Holland: Amsterdam.
[18] Solow, Robert M. (1956), “A Contribution to the Theory of Growth,” Quarterly Journal of
Economics 70:1, 65-94.
[19] Stokey, Nancy L. (1991), “Human Capital, Product Quality, and Growth,” Quarterly Journal
of Economics 106:2, 587-616.
[20] Ventura, Jaume (1997), “Growth and Interdependence,” Quarterly Journal of Economics
112:1, 57-84.
[21] Young, Alwyn (1993), “Invention and Bounded Learning by Doing,”Journal of Political Economy 101:3, 443-472.
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